A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space
Abstract
We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the AlbertiMarchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactlysupported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 DOI:
 10.48550/arXiv.2005.02924
 arXiv:
 arXiv:2005.02924
 Bibcode:
 2020arXiv200502924D
 Keywords:

 Mathematics  Functional Analysis;
 53C23;
 46E35;
 26B05
 EPrint:
 8 pages